Inverse problems for discrete heat equations and random walks for a class of graphs

Abstract

We study the inverse problem of determining a finite weighted graph (X, E) from the source-to-solution map on a vertex subset B ⊂ X for heat equations on graphs, where the time variable can be either discrete or continuous. We prove that this problem is equivalent to the discrete version of the inverse interior spectral problem, provided that there does not exist a nonzero eigenfunction of the weighted graph Laplacian vanishing identically on B. In particular, we consider inverse problems for discrete-time random walks on finite graphs. We show that under the Two-Points Condition, the graph structure and the transition matrix of the random walk can be uniquely recovered from the distributions of the first passing times on B, or from the observation on B of one realization of the random walk.

Inverse problems for locally perturbed lattices -- Discrete Hamiltonian and quantum graph

“…Namely, if (− ∆ Γ0 − λ)û = 0 in Ω i and û = ∂ ν û = 0 on ∂ Γ0 Ω i , then û = 0 in Ω i . This claim also holds for − ∆ Γ0 + q(v) with any potential q, see Lemma 3.5 in [5] or Lemma 2.4 in [6].…”

“…This indicates that û is a solution of the equation (5.37) on Ω k , ∂ Γ0 Ω k , E 0 satisfying simultaneously the Dirichlet and Neumann boundary conditions. Hence û vanishes everywhere by Lemma 2.4 in [6], provided that the subgraph Ω k , ∂ Γ0 Ω k , E 0 satisfies the assumptions (C-1) and (C-2).…”

“…By Lemma 2.4 of [6], the eigenfunctions {φ l | ∂G } l∈L k are linearly independent on ∂G, hence the rank of Q k is simply L k . When the eigenvalue λ k is simple, the matrix Q k determines φ k | ∂G up to the sign.…”